Let $X$ be a projective variety admitting a polarized (or more generally, int-amplified) endomorphism. We show: there are only finitely many contractible extremal rays; and when $X$ is $\mathbb{Q}$-factorial normal, every minimal model program is equivariant relative to the monoid $SEnd(X)$ of all surjective endomorphisms, up to finite index. Further, when $X$ is rationally connected and smooth, we show: there is a finite-index submonoid $G$ of $SEnd(X)$ such that $G$ acts via pullback as diagonal (and hence commutative) matrices on the Neron-Severi group; the full automorphisms group $Aut(X)$ has finitely many connected components; and every amplified endomorphism is int-amplified.

中文翻译：

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允许极化自同态的射影簇的所有自同态的半群结构

令 $X$ 是一个射影变体，承认极化（或更一般地，int-amplified）自同态。我们证明：只有有限多个可收缩的极值射线；当 $X$ 是 $\mathbb{Q}$-factorial normal 时，每个最小模型程序相对于所有满射自同态的幺半群 $SEnd(X)$ 是等变的，直到有限索引。此外，当 $X$ 是有理连接且平滑时，我们表明：存在 $SEnd(X)$ 的有限指数子幺半群 $G$，使得 $G$ 通过回拉作为对角（因此可交换）矩阵在Neron-Severi 组；完全自同构群 $Aut(X)$ 有有限多个连通分量；并且每个放大的内同态都是内部放大的。